O11 | Macro Paper Warehouse

Market Regulation, Cycles, and Growth Dynamics in a Monetary Union

Mon, 01 Jan 0001 00:00:00 +0000

This paper develops a two-country currency union DSGE model with endogenous TFP growth and product and labor market frictions to assess how cross-country differences in market regulation affect long-run growth and business cycle dynamics. The central insight is that with endogenous growth, there is no reason to expect real income convergence within a monetary union: large shocks can lead to permanent changes in output and the real exchange rate through their effect on endogenous TFP, lifting the standard dichotomy between cycles and growth. Less regulated economies tend to have higher trend growth and recover faster from negative shocks because their institutional environment is more conducive to innovation and reallocation. Applied to the euro area financial and sovereign debt crisis, the model is consistent with the observed divergence of output and TFP paths between Northern and Southern member states, with the less reform-friendly Southern members experiencing higher inflation, lower employment, and disappointing TFP growth.

Summary of a forthcoming paper, AI-assisted and human-reviewed. See the linked original for the authoritative claims and full conditions.

In depth

Q1. Why does endogenous growth break the convergence prediction?

With endogenous TFP growth, there is no reason to expect real income convergence within a monetary union because TFP growth depends on the institutional environment—including product and labor market regulations—which differs persistently across countries. In standard neo-classical models, capital flows toward lower-capital countries and convergence follows. But when TFP is endogenous and depends on regulations and innovation, countries with higher regulations face permanently lower TFP growth rates, and the absence of an exchange rate instrument prevents the usual adjustment mechanism from operating. The model thus provides a structural account of the non-convergence documented empirically for the euro area since 1999.

Q2. How do product and labor market regulations affect growth and cycle dynamics?

Product and labor market regulations affect both long-run trend growth (through their effect on steady-state innovation and TFP) and short-run dynamics (through their effect on how quickly economies adjust to shocks via factor reallocation). The paper documents empirically that less regulated euro area economies have higher R&D intensity and TFP growth rates. In the model, higher product market regulation reduces the incentive for firms to innovate and enter, while higher labor market regulation slows the reallocation of workers from declining to expanding sectors following a shock.

Q3. How do temporary shocks produce permanent output effects?

Temporary shocks—such as the risk premium shocks experienced by euro area countries during the financial and sovereign debt crisis—can lead to permanent reductions in the level of output and TFP through their effect on endogenous innovation and capital accumulation, producing hysteresis without any permanent shock to fundamentals. This mechanism lifts the standard dichotomy between cycles and growth: temporary financial disruptions that reduce investment and employment also reduce R&D and innovation, which lowers TFP permanently. The model thus provides a structural account of the ‘secular stagnation’ concerns following the euro area crisis.

Q4. What does the application to the euro area crisis show?

Applied to the euro area financial and sovereign debt crisis, the model is consistent with the observed divergence between Northern and Southern member states: the asymmetric risk premium shock hits less regulated Northern economies (which recover faster) and more regulated Southern economies (where output and TFP appear permanently lower) differently due to their different institutional environments. The model predicts that the divergence in output and TFP paths between Germany/France (back to pre-crisis trend) and Spain/Italy (on permanently lower paths) is consistent with the role of product and labor market regulation in mediating shock propagation, complementing the exchange rate inflexibility channel in standard currency union analyses.

Key concepts

endogenous TFP growth : TFP growth that depends on the institutional environment (product and labor market regulations) and on innovation decisions; key departure from standard DSGE models; breaks the cycle-growth dichotomy by allowing temporary shocks to permanently affect TFP levels. product market regulation (PMR) : regulations governing market entry, competition, and firm behavior in the product market; modeled here as affecting the incentive to innovate and enter new markets, thereby shaping steady-state TFP growth. labor market regulation (LMR) : regulations governing hiring, firing, and wage determination; modeled here as affecting the speed of labor reallocation following shocks, thereby shaping business cycle dynamics and recovery speed in the currency union. hysteresis : the persistence of shock effects on the long-run level of output or TFP beyond the duration of the shock itself; arises here through the effect of temporary demand contractions on endogenous innovation and TFP accumulation.

Structural Change, Land Use and Urban Expansion

Mon, 01 Jan 0001 00:00:00 +0000

This paper asks how cities grow in the process of structural transformation — specifically, whether urban expansion occurs at the intensive margin (higher density within a fixed area) or the extensive margin (larger area). The authors document and explain a persistent decline in urban density in France since 1870, and develop a spatial general equilibrium model in which endogenous land use — land allocated either to agriculture or housing — is the key mechanism linking structural change to urban sprawl.

The central empirical fact is striking: between 1870 and 2015, the area of the 100 largest French cities increased by a factor of roughly 30, while their population grew by only a factor of about 4, implying that average urban density fell by a factor of roughly 8. This density decline was fastest over 1950–1975, coinciding with the acceleration of structural change (France’s rural exodus). Since the mid-nineteenth century, approximately 15% of French land has been reallocated away from agricultural use — more than the total artificially-used land in France today (about 9%).

The theoretical mechanism operates through the opportunity cost of urban expansion. Agricultural land at the urban fringe must earn its marginal product in the rural sector; this agricultural rent pins down the cost of converting land to urban use. When agricultural productivity is low, farmland is expensive relative to income (the “food problem”), households devote large shares of resources to food, and cities remain small in area and very dense. As agricultural productivity rises — the engine of structural change — workers leave rural areas, farmland values fall relative to income, and cities can expand cheaply at their fringes. Simultaneously, richer households spend more on housing. Both forces cause urban area to grow faster than urban population, generating a sustained decline in average density.

The model also predicts a “hockey-stick” path for housing prices: during structural change, the extensive margin expansion of cities limits the rise in urban land rents despite growing housing demand. Once the reallocation of workers and land out of agriculture slows, urban land values must adjust upward rapidly, producing the pattern documented by Knoll et al. (2017) — relatively flat housing prices until roughly the 1950s, then steep increases.

The model is a multi-city, multi-sector spatial equilibrium framework with non-homothetic CES preferences (including a subsistence requirement for the agricultural good), endogenous city fringes determined by land market clearing between agricultural and residential uses, and a monocentric commuting structure with endogenous commuting speed (workers adopt faster modes as wages rise). The model is calibrated to French historical data spanning 1840–2015, with 20 regions whose sectoral productivities are estimated to match regional urban populations and local farmland prices.

Quantitatively, the calibrated model accounts for approximately 70% of the increase in urban area since 1870, most of the decline in average urban density (the factor-of-8 fall), about half of the rise in real housing prices, and most of the reallocation of land values from agricultural to urban. Cross-sectional evidence confirms a core prediction: cities surrounded by more expensive farmland are denser, with an IV-estimated elasticity of urban density with respect to farmland prices of approximately 0.3 (a 10% increase in farmland prices raises urban density by about 3%), consistent with the model’s counterpart. Scope conditions include the focus on France as a single country case, reliance on a monocentric urban structure, and the abstraction from within-urban-sector reallocation (manufacturing to services).

Q: What is the central stylized fact motivating the paper? A: Between 1870 and 2015, the area of the 100 largest French cities increased by a factor of roughly 30, while their total population grew by a factor of about 4, so average urban density fell by a factor of roughly 8. This density decline was most rapid over 1950–1975, coinciding with France’s peak rural exodus, and has barely fallen since — tracking the slowdown of structural change. This pattern is not unique to France; Angel et al. (2010) document persistent urban density decline on a global scale.

Q: What is the paper’s key theoretical mechanism linking structural change to urban sprawl? A: The rental price of agricultural land at the urban fringe is the opportunity cost of expanding the city into surrounding farmland. When agricultural productivity is low, farmland is expensive relative to income, keeping cities small and dense. As productivity rises and workers migrate to cities, the value of agricultural land falls relative to income, reducing the cost of urban expansion at the fringe. Richer households also devote a larger share of spending to housing, reinforcing the demand for space. These two channels together cause city area to grow faster than city population, generating a sustained decline in average density — even without any improvement in commuting technology.

Q: How does the paper distinguish between the structural change channel and the commuting cost channel? A: The model contains both channels: structural change (falling agricultural land values at the fringe) and falling effective commuting costs (rising wages lead workers to adopt faster commuting modes, a wage elasticity of commuting speed calibrated from survey data). Counterfactuals show that without structural change (rural productivity growth set to 4% of baseline), the model cannot replicate the observed density decline. Without faster commutes (setting the income elasticity of commuting speed to unity), the model predicts only about 30% of the baseline density decline. Both channels are necessary; their combined effect exceeds the sum of parts because structural change raises wages, which in turn amplifies the commuting speed mechanism.

Q: How do the two channels differ in their spatial imprint within cities? A: Structural change adds new low-density settlements at the urban fringe, so suburban density falls more than average density — the center is relatively less affected. Faster commuting modes, by contrast, induce suburbanization: workers relocate from the center outward, so central density falls more than average density. For Paris, historical data show that central density fell less than average urban density, which is consistent with both mechanisms operating simultaneously — the commuting channel pushing central density down more, but the structural change channel adding fringe expansion that affects suburban density more.

Q: What is the empirical evidence on the cross-sectional farmland price prediction? A: Using data on local farmland transaction prices from the French Ministry of Agriculture at the “Petite Region Agricole” level (over 700 areas), the authors show that cities surrounded by more expensive farmland are denser. A binned scatter plot across 200 French cities shows that moving from the first to last decile of farmland prices raises density by about one third — an effect comparable in magnitude to an increase in population from roughly 25,000 (3rd decile) to 150,000 (9th decile). To address endogeneity (productive cities may inflate nearby farmland prices), the authors instrument farmland prices with soil quality characteristics; the IV elasticity of urban density with respect to farmland prices is approximately 0.3, consistent with the model’s predicted counterpart.

Q: What does the model predict about the time path of housing prices? A: The model predicts a “hockey-stick” pattern: housing prices remain relatively flat for decades while structural change is ongoing, because cities expand cheaply at the extensive margin, absorbing growing housing demand without large rent increases. Once the reallocation of workers and land out of agriculture slows, the extensive margin ceases to buffer demand, and urban land values must rise sharply. The calibrated model accounts for about half of the observed rise in real housing prices since the mid-nineteenth century; it matches the qualitative hockey-stick pattern documented by Knoll et al. (2017) and Piketty and Zucman (2014) for France and advanced economies more broadly.

Q: What happens to the relative values of agricultural versus urban land over the period? A: Agricultural land values relative to income fall dramatically: the average value of a French agricultural field per unit of land, as a share of per capita income, was divided by a factor of 15 between 1850 and 2015. Meanwhile, urban land values rise. In 1820, agricultural land accounted for more than 70% of total housing and land wealth in France; by 2010 this share had fallen to about 3%. This reallocation of land values from rural to urban is a central prediction the model accounts for, driven by structural change reducing the scarcity premium on farmland.

Q: How is the model parameterized and calibrated? A: Preferences are non-homothetic CES with housing preference parameter gamma = 0.22, subsistence consumption for the rural good calibrated to match the 1840 agricultural employment share (about 60%), and substitution elasticity between urban and rural goods sigma = 0.8. The labor share in agriculture is alpha = 0.6. Commuting cost parameters (elasticities to wages and distance) are estimated from the French Labor Force Survey (Enquete Emploi). Region-specific sectoral productivity parameters for 20 regions (40 parameters total) are estimated to match the cross-section of urban populations and local farmland values in the base year 1870. The model is then simulated forward to 2015.

Q: What share of French land has been reallocated away from agriculture, and how does this relate to urban expansion? A: About two-thirds of French land was used for agriculture in 1840; by 2015 this fell to 52%, implying roughly 15 percentage points of French territory reallocated away from agricultural use. This 15% exceeds the total land currently under artificial use in France (about 9%). Over the more precisely measured period 1982–2015, artificialized soil increased by about 2 million hectares (3.7% of French territory), representing roughly 70% of the land converted away from agriculture over the same period. Two-thirds of land surrounding French cities is agricultural, confirming that urban expansion occurs at the expense of farmland.

Q: What are the limitations and directions for future research acknowledged by the authors? A: The model relies on a monocentric urban structure where all workers commute to a single city center, which is an approximation — commuting distance increases with residential distance to the center but less than one-for-one, suggesting workers sort into nearby jobs. The model also abstracts from within-urban-sector reallocation (the manufacturing-to-services transition), which the authors conjecture matters for the cross-section of cities in recent times. Finally, the model cannot fully replicate the steep recent rise in housing prices, which the authors attribute partly to land-use regulations constraining extensive margin growth — a policy counterfactual the general equilibrium structure is well-suited to analyze.

Q: How does the paper relate to the Ricardo/Nichols view that land values should rise with economic development? A: The traditional Ricardian view predicts that a fixed factor like land must rise in value with economic development — counterfactual given the historical data showing farmland values falling sharply relative to income. The authors reconcile this with the data by emphasizing that structural change and agricultural productivity growth reduce the scarcity of farmland even as total income grows, so farmland values fall. Urban land values do rise, but the structural change channel initially dampens this increase by facilitating extensive-margin city growth. The paper thus reconciles the Ricardian fixed-factor view with the commuting technology view (Miles and Sefton, 2020) within a unified spatial structural change framework.

Endogenous land use: In this paper’s framework, land in each region is allocated either to agricultural production or to residential use, with the margin between the two determined in equilibrium by the equality of the rental price of land at the urban fringe and the marginal product of land in the rural sector. This makes the urban-rural land boundary an endogenous object that responds to structural change.

Urban fringe (phi_k): The furthest residential location of an urban worker in city k, determined endogenously as the commuting distance at which the opportunity cost of further expansion (the agricultural land rent) equals the willingness of urban workers to pay for land. All workers beyond this fringe produce rural goods without commuting.

Structural change (in the paper’s sense): The reallocation of workers and land away from agriculture driven jointly by non-homothetic preferences with a subsistence consumption requirement for the agricultural good (demand side) and rising sectoral productivity (supply side). Structural change is the primary driver of falling farmland values and urban sprawl in the model.

Non-homothetic CES preferences: Household preferences over rural and urban goods that are not homogeneous of degree one in income, specified as a CES aggregate with a subsistence floor for the rural (agricultural) good. At low income levels, households devote large budget shares to food; as income rises, spending shifts toward urban goods and housing. This demand-side non-homotheticity is the channel through which rising income generates structural change.

Food problem (Schultz, 1953): The condition in which low agricultural productivity forces households to devote a large fraction of resources to meeting subsistence food needs, leaving little for housing expenditure. In the paper’s model, the food problem makes cities initially small and very dense; as agricultural productivity rises and the food problem relaxes, cities can expand in area.

Commuting cost function tau(l_k): Spatial frictions proportional to the worker’s distance from the city center and the urban wage, of the functional form tau(l_k) = a * w_{u,k}^{xi_w} * l_k^{xi_l}, where xi_w in (0,1) captures the endogenous adoption of faster commuting modes as wages rise. Concavity in both arguments is micro-founded by an optimizing commuting mode choice model, ensuring that the share of resources devoted to commuting falls as incomes rise.

Hockey-stick housing price path: The model’s prediction that real housing prices remain relatively flat over the period of active structural change — because city expansion at the extensive margin absorbs rising housing demand without large rent increases — before rising steeply once structural change slows and the extensive margin is exhausted. This prediction matches the empirical pattern documented by Knoll et al. (2017) for France and other advanced economies.

When Did Growth Begin? New Estimates of Productivity Growth in England from 1250 to 1870

Mon, 01 Jan 0001 00:00:00 +0000

Overview

Research Question. When did sustained productivity growth begin in England? This paper constructs new estimates of the evolution of productivity in England from 1250 to 1870, with the goal of both dating the onset of growth and using that dating to discriminate between competing theories of why growth began.

Methodological Innovation. The core challenge is that real wages over this period were heavily distorted by Malthusian population dynamics. Plague-induced population collapses (most dramatically the Black Death of 1348, which killed roughly 25% of England’s population) drove enormous swings in real wages that reflect movements along a stable labor demand curve, not changes in productivity. A naive regression of wages on labor supply is therefore inconsistent, because in a Malthusian world productivity growth induces population growth, making labor supply endogenous to productivity.

The authors address this by writing down and structurally estimating a full Malthusian model of the economy. Output is produced with fixed land and variable labor (and, in an extended model, capital) via a Cobb-Douglas production function. The labor demand curve equates the real wage to the marginal product of labor. Population growth is increasing in real per-capita income (the Malthus law of motion), capturing both preventive and positive checks. Productivity follows a random walk with drift, and the paper allows for two structural breaks in the average drift rate mu. Exogenous population shocks, modeled as infrequent, sizable plague draws from a beta distribution plus a Gaussian noise term, provide identification: plague shocks and productivity shocks generate observationally distinct dynamics – plague shocks cause an immediate population drop that gradually reverts, while productivity shocks cause an immediate wage jump followed by a slow population rise to a new steady state. The model is estimated via Bayesian Hamiltonian Monte Carlo (Stan), and structural break dates for mu are chosen by maximizing the Bayes factor (marginal likelihood) over the observed data on real wages, population, and days worked per worker.

Key Data. Real wages are from Clark (2010) unskilled building workers series. Post-1540 population is from Wrigley et al. (1997); pre-1540 population trends are from Clark (2007b) manorial records. Days worked per worker are from Humphries and Weisdorf (2019). All series are used as decadal averages.

Main Findings.

Onset of growth: 1600. Productivity growth was zero before 1600. The Bayes factor strongly favors a first structural break in mu at 1600; break dates before 1590 and after 1640 are clearly rejected.
Two-phase post-1600 growth. Between 1600 and 1810, average productivity growth was 4% per decade (posterior mean; 95% credible interval approximately 2%-6%). After 1810, productivity growth accelerated sharply to 18% per decade (95% CI approximately 12%-23%). The second break date is estimated to 1810 (the only alternative not clearly rejected is 1800).
Magnitude of productivity change. By the authors’ estimates, productivity in England was approximately 540% higher in 1850 than in 1500. This contrasts sharply with Clark’s (2010) dual-approach TFP series, which implies essentially no change over this period. The authors attribute the discrepancy to mismeasurement in Clark’s land rent series.
Productivity growth preceded the Glorious Revolution. Productivity rose by an estimated 48% between 1600 and 1680, well before the Glorious Revolution of 1688 and the English Civil War (1642-1651). This supports the view that economic change contributed to causing the bourgeois institutional reforms of the 17th century, consistent with the Marxist tradition (Hill, 1940, 1961), rather than that institutional change preceded and caused growth.
Weakness of Malthusian population force. The elasticity of population growth with respect to real income (gamma) is estimated at 0.09. Combined with a slope of the labor demand curve (alpha) of 0.53, this implies a half-life of plague-induced population dynamics of approximately 150 years. A doubling of real per-capita income stimulated population growth by only 6 percentage points per decade – indicating Malthusian forces were sufficiently weak to be overwhelmed by post-1800 productivity growth. The model implies that the post-1810 productivity growth rate would have produced a 28-fold long-run increase in steady-state real wages even without the Demographic Transition.
Capital extension. When capital is explicitly incorporated, using rates of return on agricultural land and rent charges to infer the capital stock, results are broadly similar: productivity growth from 1600-1810 is 3% per decade and post-1810 is 14% per decade. Capital’s production function exponent is estimated at 0.18, confirming that capital accumulation explains only a modest share of growth.

Scope Conditions. All estimates are for England specifically. The model assumes competitive factor markets, a Cobb-Douglas (or CES) production function, and a log-linear Malthusian population law of motion. Results are robust to alternative wage series (farm laborers, craftsmen, Allen’s series), alternative population sources (Broadberry et al., 2015), constant-days-worked assumption, and alternative prior distributions.

Q&A

Q1: Why can’t standard OLS regression of wages on labor supply recover productivity in this setting? In a Malthusian world, productivity growth causes population growth, which in turn raises labor supply. This means labor supply and productivity are positively correlated, biasing OLS estimates. The authors demonstrate this concretely: from 1300 to 1450 (plague era), wages and labor supply moved in opposite directions along a stable labor demand curve, while after 1630 the same data points begin shifting off that curve – a pattern that OLS would confound with changes in the slope rather than shifts in the intercept.

Q2: How do the authors distinguish empirically between a plague shock and a productivity shock? The two shocks generate fundamentally different dynamics. A plague shock causes an immediate, large drop in population and a corresponding spike in wages; over time, high wages induce population growth and both wages and population gradually return to their pre-plague levels. A permanent productivity shock, by contrast, causes an immediate rise in wages with no contemporaneous population change; population then slowly rises and wages partially revert until a new, higher steady-state population is reached. The model exploits these different impulse-response signatures in the joint data on wages and population to identify the two shocks separately.

Q3: What is the Bayes factor evidence for the 1600 break date? Figure 8 in the paper shows the Bayes factor for models with different first break dates (all holding the second break at 1810). The Bayes factor rises sharply from 1580 to 1600 and falls more gradually from 1600 to 1650. Break dates before 1590 and after 1640 are clearly rejected using the standard rule of thumb that a Bayes factor of 10 constitutes strong evidence. The 1600-1810 pair of break dates yields the highest marginal likelihood of any combination considered.

Q4: How does the paper’s productivity estimate compare to Clark’s (2010) dual-approach TFP series? Clark’s series implies productivity in England was essentially unchanged between the 15th and mid-19th centuries – a result the paper argues is implausible and inconsistent with Allen’s (2005) agricultural TFP estimates (which show a 162% increase in agricultural TFP between 1500 and 1850). The authors’ baseline estimate implies productivity was approximately 540% higher in 1850 than in 1500. The authors conjecture that a key driver of the difference is mismeasurement in Clark’s land rent series, which appears essentially flat from 1250 to 1600 despite enormous plague-induced swings in the land-labor ratio over this period.

Q5: What does the Malthusian model imply about “Engel’s Pause” – the apparent stagnation of real wages during early industrialization? Between 1730 and 1800, real wages fell slightly despite what the model estimates to be substantial productivity growth. The conventional explanation is that the gains from early industrialization accrued to capitalists rather than workers. The authors offer an alternative Malthusian explanation: England’s population grew rapidly over this period, and in the Malthusian model this population growth depressed wages relative to productivity. The authors do not reject the distributional explanation but show that Malthusian forces alone are sufficient to explain the wage-productivity divergence.

Q6: How quantitatively important are days worked (the Industrious Revolution) for the productivity estimates? The authors find that their productivity estimates are largely insensitive to whether the Humphries-Weisdorf (2019) days-worked series or a constant-days assumption is used. The qualitative pattern – zero growth before 1600, modest growth 1600-1810, rapid acceleration post-1810 – and the quantitative magnitudes remain similar. What does change is the estimated slope of the labor demand curve alpha: assuming constant days makes the labor demand curve steeper. This robustness is reassuring given that the Industrious Revolution is a contested empirical phenomenon.

Q7: What does the model imply about the speed of Malthusian population dynamics, and how does this compare to prior estimates? The estimated elasticity of population growth to real income gamma = 0.09, combined with alpha = 0.53, implies a half-life of population dynamics of approximately 150 years. This is consistent with but lies between prior structural estimates: Lee and Anderson (2002) find a half-life of 107 years, and Crafts and Mills (2009) find 431 years. All estimates agree that Malthusian dynamics in England were slow relative to the conceptual ideal of rapid subsistence convergence.

Q8: Can the model explain the post-1750 population explosion without invoking the Demographic Transition? Yes. The authors simulate predicted population paths from 1740 to 1860 taking real wages and days worked as given and using their estimated gamma and alpha. Despite the weak Malthusian population force, the model can explain the vast majority of the observed population growth from 6 million in 1740 to nearly 20 million in 1860 (10.4% per decade). The key mechanism is that days worked increased substantially over this period, raising per-capita income well above what real wages alone would suggest.

Q9: How does incorporating capital change the productivity estimates? In the capital-augmented model, the capital stock is inferred from rates of return on agricultural land and rent charges (Clark 2002, 2010). The capital exponent beta is estimated at 0.18, indicating a modest role for capital in pre-industrial England. Average productivity growth from 1600-1810 falls from 4% to 3% per decade, and post-1810 growth falls from 18% to 14% per decade. The authors conclude that the vast majority of growth from 1600 to 1870 cannot be attributed to capital accumulation. From 1600 to 1860, the estimated capital stock grew by a factor of five (8% per decade).

Q10: What theories of the onset of growth are consistent vs. inconsistent with the authors’ timing evidence? Inconsistent: The North-Weingast (1989) view that the Glorious Revolution of 1688 was the key institutional trigger, since productivity had already risen 48% between 1600 and 1680. Also inconsistent: gradual-growth theories (Kremer 1993, Galor-Weil 2000) in which there is no discrete acceleration. Consistent: Marxist accounts (Hill 1940, 1961) that economic change drove 17th-century institutional change; Acemoglu-Johnson-Robinson (2005) accounts linking Atlantic trade enrichment to the demand for secure property rights (timing broadly consistent, though growth rates do not visibly accelerate after the Civil War or Glorious Revolution); cultural-change accounts (Mokyr, McCloskey) tracing the onset of growth to the spread of literacy and scientific rationalism around 1600; Allen’s (2009a) directed-technical-change theory linking 17th-century wage growth to the later profitability of labor-saving innovation in the Industrial Revolution.

Q11: What does the model imply about the long-run real wage consequences of post-1810 productivity growth, even counterfactually assuming Malthusian forces persisted? The steady-state real wage in the Malthusian model is w-bar = mu/(alpha*gamma) minus subsistence-related terms. For mu = 0.018 (the post-1810 estimate), this formula implies a long-run real wage 28 times higher than the steady state under zero productivity growth. In other words, even if the Demographic Transition had not occurred and birth and death rates had remained sensitive to income, post-1810 productivity growth was fast enough relative to the weak Malthusian force to generate substantial sustained rises in living standards.

Key Concepts

Labor demand curve (in the paper’s sense). The equilibrium relationship between real wages and labor supply derived from competitive profit maximization by landowners facing a fixed land endowment: w_t = phi - alpha*l_t + a_t. Productivity is identified as shifts in this curve across time periods. The slope alpha is not simply the land share under a CES production function but equals one minus the labor share divided by the elasticity of substitution between labor and land.

Malthusian population force. The feedback mechanism by which higher real wages induce faster population growth, expanding labor supply and pushing wages back toward a steady state. Its speed is governed jointly by gamma (elasticity of population growth with respect to income) and alpha (slope of the labor demand curve); the half-life of wage/population dynamics after a shock equals log(0.5)/log(1 - alpha*gamma). In the paper’s estimates, this force was sufficiently weak (half-life approximately 150 years) that post-1800 productivity growth overwhelmed it.

Plague shock (xi_1t). An infrequent, large, exogenous negative population shock modeled as a draw from a beta distribution occurring with probability pi. Plagues are the primary source of identifying variation for the pre-1600 period: they generate movements along a stable labor demand curve and allow the slope alpha and the (lack of) productivity trend to be separately identified from labor demand shifts.

Structural break in average productivity growth (mu). The drift parameter in the random-walk model for the permanent component of productivity. The paper allows two breaks in mu, with break dates chosen to maximize the marginal likelihood (Bayes factor). The best-fitting breaks are at 1600 (zero to 4% per decade) and 1810 (4% to 18% per decade).

Permanent vs. transitory productivity component. Productivity is decomposed into a permanent component a-tilde_t (random walk with drift, sigma_epsilon1) and a transitory component epsilon_2t (iid noise, sigma_epsilon2). The paper reports and interprets the permanent component as the meaningful measure of underlying technological change; transitory shocks are treated as measurement error and short-run fluctuations.

Industrious Revolution. The hypothesized long-run increase in days worked per worker in England, associated with de Vries (1994, 2008). The paper uses Humphries-Weisdorf (2019) estimates showing a sharp drop after the Black Death followed by a sustained rise from 1350 onward. A key robustness result is that the paper’s productivity estimates are insensitive to whether this Industrious Revolution is assumed to have occurred.

Bayes factor (model selection). The ratio of marginal likelihoods p(y|M_t)/p(y|M_t’) for two competing models, used here to select structural break dates for mu. A factor of 10 is treated as strong evidence. The bridge sampling method of Gronau, Singmann, and Wagenmakers (2020) is used to compute marginal likelihoods.